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\title[Robust Optimization]{Robust optimization (stability) of dynamical system}
\subtitle{closed-loop robust control approach}

\author[HU Jun]{Jun HU}
\institute{Orange Labs}
\date{\today}


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\frametitle{Content}
\tableofcontents
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\section{Case study}
\subsection{Dynamical system and uncertainty}
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\frametitle{Case study}


\begin{columns}
\column{0.7\textwidth}
Types of uncertainty:

\begin{itemize}
\item Deterministic: model error (unknown, simplification, etc.,)
\item Stochastic: call demands (social activity)
\begin{itemize}
\item Periodic demands during daytime.
\item Unpredictable hotspots. 
\item The call demands in the network are not homogeneous
\end{itemize}
\end{itemize}

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\includegraphics[width=.7\textwidth]{pic/networkHeterogenous}
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\vfill
Consider SON as an interconnected dynamical system (self-organizing, self-healing etc.,)
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\section{Methods}

\subsection{Approaches in literature}

\begin{frame}
\frametitle{Approaches in literature: static}
\begin{itemize}
\item Convex counterpart approach:
\begin{itemize}
\item Transform an important class of mini-max optimization problems into tractable convex optimization problems
\item Find a mini-max solution which minimizes the worst scenario.  
\item Conservative and static
\end{itemize} 

\vitem Stochastic optimization:
\begin{itemize}
\item Regard the uncertain parameter in the optimization problem as a random variable for which a given probability distribution is assumed. In the corresponding chance constrained formulation, the probability of a constraint violation is asked to be below a given confidence probability.
\item Find a solution which minimizes the current cost and expected cost for possible scenarios. 
\end{itemize} 

\end{itemize}
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\begin{frame}
\frametitle{Approaches in literature: dynamic}
\begin{itemize}
\item Recoverable robustness
\begin{itemize}
\item Similar to two-stage stochastic optimization, 
\item Find a pair $(x,A)$ where $x$ is solution, $A$ is a repair algorithm. When the uncertainty reveals, new solution can be found by applying $A(x)$. 
\item Converge slowly.
\end{itemize}
\vitem Online anticipatory stochastic optimization: \cite{Bent2004}
\begin{itemize}
\item Given a stochastic process to sample the future scenarios and an offline algorithm. 
\item At each time stamp, the offline algorithm carrying out a solution which minimizes the sampled future scenarios. 
\end{itemize}

\vitem Control theory -- closed-loop control and robust stability
\begin{itemize}
\item Feedback controller  
\item FeedForwarding MPC (Model Predictive Control) controller
\end{itemize}
\end{itemize}
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%\item The distribution of uncertainty is known or can be estimated through historical data. 
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\section{Control vs Optimization}
\subsection{Control theory approach}

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\frametitle{Feedback control}
\begin{itemize}
\item Dynamical system -- consider the network as a dynamical system, the system can be illustrated as: 
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\vitem Control vs. Optimization: consider the controller as decision variable of optimization problem, desired objectives are constraints on controlled closed-loop system. 

\vitem Crucial difficult: nonlinear dynamic system linearization
\end{itemize}
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\frametitle{Online anticipatory stochastic optimization}
\begin{itemize}
\item Condition: the knowing the history data of stochastic uncertainty or the distribution of such events 
\vitem 
\end{itemize}
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\begin{frame}
\frametitle{Solutions pool}
\begin{itemize}
\item Solutions pool maintained by evolutionary algorithm
\begin{itemize}
\vitem Each solution is robust bounded by mean-variance scenarios set
\vitem The population is classified into three categories: \cite{Hatzakis2006eampc}
\begin{itemize}
\vitem[Front:] Coverage to current terrain
\item[Cruft:] Maintain the diversity (handle unforecasted change)
\item[Prediction:] Predict new optimum (handle forecasted change)
\end{itemize}
\end{itemize}
\vitem Distance measure between solutions
\begin{itemize}
\vitem Distance is measured by the modification cost between two solutions
\vitem Update the solutions in solutions pool by the distances among them
\end{itemize}
\end{itemize}
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\frametitle{References}
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